Subject: Question about manifolds
Date: Tue, 19 Feb 2002 14:39:14 -0500
From: Gregory Arone <zga2m@weyl.math.virginia.edu>
To: dmd1@lehigh.edu

Dear Don

Could you post this for me?

Thanks, Greg.

Let M be a manifold, N an embedded submanifold
(for simplicity, manifold = compact manifold without
boundary). Consider the space M-N (the complement
of N in M). I would like to "compactify" this space, without
changing the homotopy type, by "glueing" the sphere
bundle of the normal bundle of N to M-N. Such
a compactification, once constructed, will be denoted
cl(M-N).

One way to construct cl(M-N) is to choose an open tubular
neighborhood of N in M, and take the complement of
this neighborhood. My question is: is there a more
canonical way to do it?

Ideally, I would like a construction for cl(M-N) that does
not involve a choice of anything like a Riemannian
metric, nor  a choice of a complement of the tangent
bundle of N in the tangent bundle of M. The construction
should be as functorial as one could reasonably expect.
E.g., an embedding f:M--> M' would induce a map
cl(M-N)-->cl(M'-f(N))

To summarize, my questions are:

Is such a construction possible?

Is it described in the literature?

With many thanks
Greg